It is given that AA is a 6×9 matrix and Nullity \(\displaystyle{\left({A}^{{T}}\right)}={2}\). Let rank(A)=k. Then we know that rank \(\displaystyle{\left({A}^{{T}}\right)}={k}\). By Rank-nullity theorem we have

rank(A)+Nullity(A)=9andrank(AT)+Nullity(AT)=6

It follows that

rank \(\displaystyle{\left({A}^{{T}}\right)}={6}−{2}={4} ⟹ {k}={4}\)

Therefore we get

Nullity(A)=9−k=9−4=5

Hence the given statement is False.

rank(A)+Nullity(A)=9andrank(AT)+Nullity(AT)=6

It follows that

rank \(\displaystyle{\left({A}^{{T}}\right)}={6}−{2}={4} ⟹ {k}={4}\)

Therefore we get

Nullity(A)=9−k=9−4=5

Hence the given statement is False.